Question

A subcommittee of six is to be selected from a committee containing 10 democrats and 12 republicans. In how many ways can at least 1 democracy be selected for the subcommittee?

Answer 1

the number of ways to select at least 1 democrat in the subcommittee is 69,486 ways

Explanation:

Given;

number of the subcommittee, = 6

number of democrats = 10

number of republicans, = 12

The number of ways to select at least 1 democrat in the subcommittee is calculated as follows;

Let D represent Democrats

let R represent Republicans

= (1D & 5R) or (2D & 4R) or (3D & 3R) or (4D & 2R) or (5D & 1R) or (6D)

= 10C₁ x 12C₅ + 10C₂ x 12C₄ + 10C₃ x 12C₃ + 10C₄ x 12C₂ + 10C₅ x 12C₁ + 10C₆

`=( (10!)/(9!1!) * (12!)/(7!5!) )+ ((10!)/(8!2!) * (12!)/(8!4!))+ ((10!)/(7!3!) * (12!)/(9!3!))+ ((10!)/(6!4!) * (12!)/(10!2!))+ ((10!)/(5!5!) * (12!)/(11!1!)) \\\\ +((10!)/(4!6!))\\\\= (7,920) + (17,820) + (26,400) + (13,860)+ (3,276) + (210)\\\\= 69,486 \ ways`

Therefore, the number of ways to select at least 1 democrat in the subcommittee is 69,486 ways